# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

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## Mathematical Control & Related Fields

March 2021 , Volume 11 , Issue 1

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2021, 11(1): 1-22 doi: 10.3934/mcrf.2020024 +[Abstract](1286) +[HTML](435) +[PDF](517.15KB)
Abstract:

In this paper we consider the optimal dividend problem for an insurance company whose surplus follows a classical Cramér-Lundberg process with a feature of self-exciting. A Hawkes process is applied so that the occurrence of a jump in the claims triggers more sequent jumps. We show that the optimal value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a given boundary condition and declare its concavity. We introduce a barrier curve strategy and verify its optimality. Finally, some numerical results are exhibited.

2021, 11(1): 23-46 doi: 10.3934/mcrf.2020025 +[Abstract](1461) +[HTML](328) +[PDF](556.27KB)
Abstract:

This paper considers a class of linear-quadratic-Gaussian (LQG) mean-field games (MFGs) with partial observation structure for individual agents. Unlike other literature, there are some special features in our formulation. First, the individual state is driven by some common-noise due to the external factor and the state-average thus becomes a random process instead of a deterministic quantity. Second, the sensor function of individual observation depends on state-average thus the agents are coupled in triple manner: not only in their states and cost functionals, but also through their observation mechanism. The decentralized strategies for individual agents are derived by the Kalman filtering and separation principle. The consistency condition is obtained which is equivalent to the wellposedness of some forward-backward stochastic differential equation (FBSDE) driven by common noise. Finally, the related \begin{document}$\epsilon$\end{document}-Nash equilibrium property is verified.

2021, 11(1): 47-71 doi: 10.3934/mcrf.2020026 +[Abstract](1281) +[HTML](281) +[PDF](395.83KB)
Abstract:

This paper is concerned with mean-field stochastic linear-quadratic (MF-SLQ, for short) optimal control problems with deterministic coefficients. The notion of weak closed-loop optimal strategy is introduced. It is shown that the open-loop solvability is equivalent to the existence of a weak closed-loop optimal strategy. Moreover, when open-loop optimal controls exist, there is at least one of them admitting a state feedback representation, which is the outcome of a weak closed-loop optimal strategy. Finally, an example is presented to illustrate the procedure for finding weak closed-loop optimal strategies.

2021, 11(1): 73-94 doi: 10.3934/mcrf.2020028 +[Abstract](1243) +[HTML](285) +[PDF](342.43KB)
Abstract:

This paper is concerned with a kind of nonzero-sum differential game of backward doubly stochastic system with delay, in which the state dynamics follows a delayed backward doubly stochastic differential equation (SDE). To deal with the above game problem, it is natural to involve the adjoint equation, which is a kind of anticipated forward doubly SDE. We give the existence and uniqueness of solutions to delayed backward doubly SDE and anticipated forward doubly SDE. We establish a necessary condition in the form of maximum principle with Pontryagin's type for open-loop Nash equilibrium point of this type of game, and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a nonzero-sum differential game of linear-quadratic backward doubly stochastic system with delay.

2021, 11(1): 95-117 doi: 10.3934/mcrf.2020029 +[Abstract](1301) +[HTML](258) +[PDF](1685.06KB)
Abstract:

We study the robust output regulation of linear boundary control systems by constructing extended systems. The extended systems are established based on solving static differential equations under two new conditions. We first consider the abstract setting and present finite-dimensional reduced order controllers. The controller design is then used for particular PDE models: high-dimensional parabolic equations and beam equations with Kelvin-Voigt damping. Numerical examples will be presented using Finite Element Method.

2021, 11(1): 119-141 doi: 10.3934/mcrf.2020030 +[Abstract](1174) +[HTML](255) +[PDF](370.14KB)
Abstract:

In the present work, a nonlocal optimal design model has been considered as an approximation of the corresponding classical or local optimal design problem. The new model is driven by the nonlocal \begin{document}$p$\end{document}-Laplacian equation, the design is the diffusion coefficient and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove the existence of an optimal design for the new model. This work is complemented by showing that the limit of the nonlocal \begin{document}$p$\end{document}-Laplacian state equation converges towards the corresponding local problem. Also, as in the paper by F. Andrés and J. Muñoz [J. Math. Anal. Appl. 429:288– 310], the convergence of the nonlocal optimal design problem toward the local version is studied. This task is successfully performed in two different cases: when the cost to minimize is the compliance functional, and when an additional nonlocal constraint on the design is assumed.

2021, 11(1): 143-167 doi: 10.3934/mcrf.2020031 +[Abstract](1110) +[HTML](257) +[PDF](416.46KB)
Abstract:

The objective of the paper is to contribute to the theory of error-based control systems on Riemannian manifolds. The present study focuses on system where the control field influences the covariant derivative of a control path. In order to define error terms in such systems, it is necessary to compare tangent vectors at different points using parallel transport and to understand how the covariant derivative of a vector field along a path changes after such field gets parallely transported to a different curve. It turns out that such analysis relies on a specific map, termed principal pushforward map. The present paper aims at contributing to the algebraic theory of the principal pushforward map and of its relationship with the curvature endomorphism of a state manifold.

2021, 11(1): 169-188 doi: 10.3934/mcrf.2020032 +[Abstract](1433) +[HTML](273) +[PDF](395.6KB)
Abstract:

The paper is devoted to analyze the connection between turnpike phenomena and strict dissipativity properties for continuous-time finite dimensional linear quadratic optimal control problems. We characterize strict dissipativity properties of the dynamics in terms of the system matrices related to the linear quadratic problem. These characterizations then lead to new necessary conditions for the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities. One of the key novelty of these results is the possibility to encompass the presence of state and input constraints.

2021, 11(1): 189-209 doi: 10.3934/mcrf.2020033 +[Abstract](1897) +[HTML](250) +[PDF](562.18KB)
Abstract:

In this paper, we study optimal control problems for nonlinear fractional order boundary value problems on a star graph, where the fractional derivative is described in the Caputo sense. The adjoint state and the optimality system are derived for fractional optimal control problem (FOCP) by using the Lagrange multiplier method. Then, the existence and uniqueness of solution of the adjoint equation is proved by means of the Banach contraction principle. We also present a numerical method to find the approximate solution of the resulting optimality system. In the proposed method, the \begin{document}$L2$\end{document} scheme and the Grünwald-Letnikov formula is used for the approximation of the Caputo fractional derivative and the right Riemann-Liouville fractional derivative, respectively, which converts the optimality system into a system of linear algebraic equations. Two examples are provided to demonstrate the feasibility of the numerical method.

2021, 11(1): 211-236 doi: 10.3934/mcrf.2020034 +[Abstract](978) +[HTML](233) +[PDF](533.44KB)
Abstract:

In the paper, the problems of controllability and approximate controllability are studied for the control system \begin{document}$w_t = w_{xx}$\end{document}, \begin{document}$w_x(0,\cdot) = u$\end{document}, \begin{document}$x>0$\end{document}, \begin{document}$t\in(0,T)$\end{document}, where \begin{document}$u\in L^\infty(0,T)$\end{document} is a control. It is proved that each initial state of the system is approximately controllable to each target state in a given time \begin{document}$T$\end{document}. A necessary and sufficient condition for controllability in a given time \begin{document}$T$\end{document} is obtained in terms of solvability of a Markov power moment problem. It is also shown that there is no initial state which is null-controllable in a given time \begin{document}$T$\end{document}. Orthogonal bases are constructed in \begin{document}$H^1$\end{document} and \begin{document}$H_1$\end{document}. Using these bases, numerical solutions to the approximate controllability problem are obtained. The results are illustrated by examples.

2020 Impact Factor: 1.284
5 Year Impact Factor: 1.345
2020 CiteScore: 1.9